Generalized Robertson-Walker metrics and some of their properties. II

Abstract

The Robertson-Walker (RW) metrics, of dimensionality four and signature −2, are generalized to metrics of dimensionality (n+1) and of arbitrary signature,n (> 1) being an arbitrary integer. In canonical coordinates (t, x 1,x 2, ...,x n ) these generalized Robertson-Walker (GRW) metrics are functions of the coordinatet. The following statements are proved to be equivalent: The GRW metrics are (a) expressible int-independent form, (b) of constant curvature, (c) Einstein spaces. Furthermore, there are six, and only six, classes of GRW metrics satisfying these three statements. The coordinate transformations which transform these metrics to theirt-independent form are given explicitly. Two of these classes of GRW metrics reduce, in theirt-independent form, to the same flat (generalized Minkowski) metrics, three reduce to the samet-independent metrics which are generalizations of the de Sitter space-time metric, and the last class tot-independent metrics which are generalizations of the anti-de Sitter space-time metric.